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Tumbling Asteroids Tumbling Asteroids Alan W. I]arris A4S 183-501, .Jet l’ropul.vim l.ahmtory l’clsdwl, CA, 91109 Submitted as a Note to lcarus July 31, 1993 S Manl]script pages 1 Figlll”e Suggest cd running head: ~iumbling Asteroids Send corrcspondcncc to: Alan W. 1 ]arris Al\ S’1’RAC’l’ l]urns and Safrcmov (A40n. Not, 1/, As(wtr. ,Voc. 16S, 403-411, 1973) cstimalcd the damping timescalc of rotational \vobblc for asteroids, and concluded that all askxoid rotations then knowm should bc damped to a state of plincipal-axis rotation about tbc axis of maxilnunl mmncn( of inertia, I have rc-examined this c]ucstion in the light of some more rcccnt]y dctcrmincd case.s of very slo~v rotation rates, and find that for several asteroids, the damping tinmcalc is cxpcctcd to bc considerably longer tba]l the agc of the solar systcm, implying that these objects may t’cry \vcll exhibit non-principal axis rotation: ~~obblc, or in cxt rcmc cases, the appearance of “tumbling” in space. I’crhaps most notable in this group is the asteroid 4179 ‘J’outatis, Both radar observations (Os(ro, personal corllr]lll]~icatioll) and lightcurvc observations (Barucci, Spcnccr, personal communications) .suggcst that Toutatis may indeed bc in a more complex state of rotational motion than simple principal axis rotation 1 mention several other examples of objects which might bc cxpcctcd to bc in similar s[atcs, and a couple examples of ligbtcurvc observations of such objects that appear to support that conclusicm. MOSI of our knotvlcdgc regarding the rotalion states of as(croids comes from photometric lightcur~’cs. ‘1’o date periods have been dctcrmincd for more than 600 otjccts, }’or about 100 of these, the lightcurvcs arc so fflcicntly dclailcd and cover cllough cyc]cs to support the cor[clusio[t that the rotalion is constant about the axis of rnaxill]unl moment of incttia. I’hat is, the lightcurvc is strictly periodic with a single rotation frequency (l)lus harmonics), cxccpt for ~’ariations attributable to changing aspect of illumination and vic\ving. Since it is generally bclicvcd that the rotational motion is the result of collisional proccsscs, and among smaller asteroids involves catastrophic disruptions from larger parent bodies, onc would expect that o] isinal spin states involved large amplitude wobbles, rcsolting in a complex “tumbling” t}~c of nlotion, such as appears to bc the case for the nucleus of (omct P/lla]lcy (Peale and I,issaucr 1989; Ilclton 1991). IIurns and Safronov (1973) examined tbc rate of dissipation of rotational culcrgy as a result the stress-strain within an asteroid tvhich arises from non-principal axis rotation Such motion is cxaclly analogous to the I:arth’s C%andlcr Wobble, and as with the Iiatlh, is damped duc to stress-strain hyslcrcsis willlirl the body, “1’hcy derive the follolving expression for tk time scale T of damping of the ~vobb]c to a state of principal axis rotation: T -- pQ/(pK32r2m3), where p is the rigidity of the material composing the asteroid, Q is IIIC quality factor (ratio oft hc energy contained in the oscillation to Ibc energy lost pcr cycle), p is the bulk density of the body, K32 is a numerical factor relating to IIIC irregularity of tbc body ranging from -0.01 for a ncar]y-spherical onc to -0,1 for a highly elongate or oblatc OIIC, r is tbc mean radius of the asteroid, and (o is the angular frequency of rotation, Burns and Safronov compute damping timcscalcs for 4 example asteroids, obtailting timcscalcs ranging from -lOs years to -108 years, I’hus Ihcy concluded that probably all observed asteroids arc in a state of damped rotational motion, At the time that they made tbcir analysis, the lon~cst rotaticm period “known” (McAdoo and Burns 1973) was 18.813 hours (for 5321 Icrculina; that reported value subsequently turned out to bc wro]~g), so in the contcx( of all rotatio]~s known tbcn, tbcir conclusion was quite il refutable, c~w allowing for a generous range of uncertainty of the parameters involved, most notably }(Q, It is clear ho~vcvcr, duc to the factors rb3, the damping, time could be very long for small asteroids }vith very long rotation periods, ‘1’his was pointed out by McAdoo and Burns (1974), but pro]nptly forgotten by most cvcr~’one, myself illcludcd, At the urging of S, J, Ostro (personal communicant ion), 1 rcconsidcrcd the timcscalcs of wobble damping for tbc nolv much larger data set of asteroid rotations, which include several examples of many-day rotation periods, l’hc case in point \vas tbc asteroid 4179 “1’outatis, which was imaged on many days with radar by Ostro and his collaborators, and was observed pbotoclcctrically \’cry cxtcnsivc]y ot’cr t}vo months from Dcccmbcr 1992 through January ] 9°3 (]]arLICCl, Spcnccr, persona] co]]~][illllicatiolls), Prclimiltary cxamina[ion of both the radar and optical ciala indicate that ‘1 ‘outatis is a very irregular object ~~itb a very long rotation “period”, and suggest that I’outatis’ rotatio]m] motion is more complex than simple principal axis rotation A quick substitution of the physical parameters appropriate for ‘1’outatis (rotation period ~-7.5 days, r -2 kn~, K32 - 0.05), and adopting Bum’ and x. (,,.1, ? Safronov’s WIUCS of all the other parameters, YICICJS a timcscale of -1.5 x 10’2 years. ‘1’hus onc should cxpwl that ‘J’outatis’ rotational motion is no[ simple, even if it is as old as the agc of the solar systcm. ~onsidcril]g that asteroids only a fcw kilometers in diameter arc cxpcctcd to have a collisional lifctin~cs of only a small fraction of the agc of the solar systcm (probably <108 years), it is even mote likely that ‘1’outatis’ rotational \vobblc is undamped, Iollowin~ the rcalimtion that there arc now cases where the wobble damping time is cxpcctcd to bc long comprrrcct to (1IC ages of the asteroids, 1 rc-cvaluatccl the above ccplation in greater detail, using more current estimates for the other parameters. I C11OSC p = 2.0 al)d pQ = S x 10’2, based on estimates of these parameters for I’hobos, the only small body in the relevant siz.c rang,c for which \vc have estimates of these properties. ‘1’hc cstinlatc of jIQ for I’}~obos is based on the observed eccentricity of its orbit about Mars, and a plausib]c past orbital history under tidal evolution (Yodcr 1982). Burns and Safronov chose a value of itQ = 3 x 10]4, which would bc appropriate for a solid, non-porous rock, and is likely too high, but was appropriately chosen to obtain a firm upper limit in their work Peale and I,issaucr (1989) sugucst that p c: 10’0 and Q <100 might bc appropriate for ~omct 12 1 Iallcy. I’hus I conclude that pQ probably lies ~vithin a factor of 10 of the assumed value of 5 x 10 , and certainly wit hin a factor of 100. ‘J’hc uncertainty in p probably dots not cxcccd a factor of 1.5, so the error bud~ct is dominated by pQ, I;vcn the range of vahm possible of K32, from 0.01 to 0.1, is hardly significant compared to the uncertainty in pQ. ‘1’hus in the relation below, I have adopted a uniform value of 0.03, even though onc could in principle c.stimatc values for each individual as[croid based on lightcurvc amplitude, Using these conslants, I derive the following relationship bctwccn rotation period P in hours, diameter D in km, and damping timcscalc ‘t in billions ofycars: }, ,. ]7 1)23T1/3, I’hc uncertainty in the constant is about a factor of 2,5, combining the unccr[ainties in all of the other parameters; that is, P = (7-40) D2’3 T} ’3. in }iig,urc J, 1 hal’c plotted solutions to the above relation for three vahrcs of ‘t on a plot of measured asteroid rotation rates vs., diarnctcrs. It has generally been claimed that most of the present asteroids, especially those s[tmllcr tl]all 100-200 km in diameter, arc not primordial, but rather arc the products of catastrophic disruptions. Such an event should “rc.set” the rotation state of tlic fragnlcnts, and especially il~ducc large \vobblc amplitudes. Rcccllt estimates of the collision energy threshold for catastrophic disruption as a function of size of the target body (lloLIscn C( al. 1991) imply that the cxpcctcd “age” of a SO km asteroid may bc – 10s years, and objects a fcw kn~ in diameter may survive only107 - . years, ‘1’hus in the figure, any object that falls below even the topmost line plotted should bc considered a candidate for non-principal axis rotation, and those below the bottom line arc almost ccrlain to bc in a state of large amplitude wobble, or “tumbling”. I have identified in the figure all of the objects that have estimated T <1 b,y. Note that the four objects, 3102 1981 QA, 4179 ‘1’outatis, 1220 Crocus, and 288 Glaukc, arc most cxtrcmc, having damping timcscalcs much longer than the agc of the solar systcm. As I have already noted, the obscrvatio]lal data on I’outatis is at least consis[cnt with, if not requiring, tumbling motion A major point of this note is to cncouragc observers to evaluate tllc availab]c data in the light of this prospect, of the other three objects, it can bc said that the cxisling data arc i]mtTcicnt to demonstrate non-principal axis rotation, but arc not inconsistent \vit}l that possibility (SCC Ilarris ef GI, 1992 for the lightcurvc of 3 102; Bin~.cl 1985 for 1220; and IIarris 1983 for 288), Moving up to the asteroids ~vith T -- 4,5 by., 887 Alinda was observed cxtcnsivcly during two apparitions, in 1969/70 and 1973/74, by Dunlap and I’aylor (1979), but again the overlap in covcragc was inadequate to make any definite statcrncnt about the prcsc]lcc or abscncc of wobble.
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